Small BGK Waves and Nonlinear Landau Damping |
| |
Authors: | Zhiwu Lin Chongchun Zeng |
| |
Institution: | 1.School of Mathematics,Georgia Institute of Technology,Atlanta,USA |
| |
Abstract: | Consider a 1D Vlasov-poisson system with a fixed ion background and periodic condition on the space variable. First, we show
that for general homogeneous equilibria, within any small neighborhood in the Sobolev space ${W^{s,p}\left( p >1 ,s <1 +\frac{1}{p}\right)}${W^{s,p}\left( p >1 ,s <1 +\frac{1}{p}\right)} of the steady distribution function, there exist nontrivial travelling wave solutions (BGK waves) with arbitrary minimal
period and traveling speed. This implies that nonlinear Landau damping is not true in
Ws,p( s < 1 +\frac1p){W^{s,p}\left( s <1 +\frac{1}{p}\right)} space for any homogeneous equilibria and any spatial period. Indeed, in a
Ws,p(s < 1 +\frac1p){W^{s,p}\left(s <1 +\frac{1}{p}\right)} neighborhood of any homogeneous state, the long time dynamics is very rich, including travelling BGK waves, unstable homogeneous
states and their possible invariant manifolds. Second, it is shown that for homogeneous equilibria satisfying Penrose’s linear
stability condition, there exist no nontrivial travelling BGK waves and unstable homogeneous states in some ${W^{s,p}\left( p >1 ,s >1 +\frac{1}{p}\right)}${W^{s,p}\left( p >1 ,s >1 +\frac{1}{p}\right)} neighborhood. Furthermore, when p = 2, we prove that there exist no nontrivial invariant structures in the ${H^{s}\left( s > \frac{3}{2}\right) }${H^{s}\left( s > \frac{3}{2}\right) } neighborhood of stable homogeneous states. These results suggest the long time dynamics in the ${W^{s,p}\left( s >1 +\frac{1}{p}\right) }${W^{s,p}\left( s >1 +\frac{1}{p}\right) } and particularly, in the ${H^{s}\left( s > \frac{3}{2}\right) }${H^{s}\left( s > \frac{3}{2}\right) } neighborhoods of a stable homogeneous state might be relatively simple. We also demonstrate that linear damping holds for
initial perturbations in very rough spaces, for a linearly stable homogeneous state. This suggests that the contrasting dynamics
in W
s, p
spaces with the critical power
s=1+\frac1p{s=1+\frac{1}{p}} is a truly nonlinear phenomena which can not be traced back to the linear level. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|